Hey! I'm David, the author of the Real-World Cryptography book. I was previously a crypto architect at O(1) Labs (working on the Mina cryptocurrency), before that I was the security lead for Diem (formerly Libra) at Novi (Facebook), and a security consultant for the Cryptography Services of NCC Group. This is my blog about cryptography and security and other related topics that I find interesting.

The socat thingy created some interest in my brain and I'm now wondering how to build a NOBUS (Nobody But Us) backdoor inside Diffie-Hellman and how to reverse it if it's not a proper NOBUS.

Ways to do that is to imagine how the DH non-prime modulus could have been generated to allow for a backdoor. For it to be a NOBUS it should not easily be factorable, but for it to allow a Pohlig-Hellman attack it must have a B-smooth order with B small enough for the adversary to compute the discrete log in a subgroup of order B.

If you go on the github repository you will see an already working proof of concept that explains each of the steps (generation, attack)

This proof of concept underlines one of the ways the malicious committer could have generated the non-prime modulus \(p = p_1 p_2\) with both \(p_i\) primes such that \(p_i - 1\) are smooth. The attack works, but I'm thinking about ways of reversing such a non-prime modulus that would disable the NOBUS property of the backdoor. Spoiler alert: Pollard's p-1 factorization algorithm.

Anyway, if you're interested in contributing to that research, or if you have any comments that could be useful, please shoot me a message =)

In the OpenSSL address implementation the hard coded 1024 bit DH p parameter was not prime. The effective cryptographic strength of a key exchange using these parameters was weaker than the one one could get by using a prime p. Moreover, since there is no indication of how these parameters were chosen, the existence of a trapdoor that makes possible for an eavesdropper to recover the shared secret from a key exchange that uses them cannot be ruled out.
A new prime modulus p parameter has been generated by Socat developer using OpenSSL dhparam command.
In addition the new parameter is 2048 bit long.

This is a pretty weird message with a Juniper feeling to it.

Socat's README tells us that you can use their free software to setup an encrypted tunnel for data transfer between two peers.

Looking at the commit logs you can see that they used a 512 bits Diffie-Hellman modulus until last year (2015) january when it was replaced with a 1024 bits one.

Socat did not work in FIPS mode because 1024 instead of 512 bit DH prime is required. Thanks to Zhigang Wang for reporting and sending a patch.

The person who pushed the commit is Gerhard Rieger who is the same person who fixed it a year later. In the comment he refers to Zhigang Wang, an Oracle employee at the time who has yet to comment on his mistake.

The new DH modulus

There are a lot of interesting things to dig into now. One of them is to check if the new parameter was generated properly.

It is a prime. Hourray! But is it enough?

It usually isn't enough. The developper claims having generated the new prime with openssl's dhparam command (openssl dhparam 2048 -C), but is it enough? Or even, is it true?

To get the order of the DH group, a simple \(p - 1\) suffice (\(p\) is the new modulus here). This is because \(p\) is prime. If it is not prime, you need to know its factorization. This is why the research on the previous non-prime modulus is slow... See Thai Duong's blogpost here, the stackexchange question here or reddit's thread.

Now the order is important, because if it's smooth (factorable into "small" primes) then active attacks (small subgroup attacks) and passive attacks (Pohlig-Hellman) become possible.

So what we can do, is to try to factor the order of this new prime.

Here's a small script I wrote that tries all the primes until... you stop it:

# the old "fake prime" dh params
dh1024_p = 0xCC17F2DC96DF59A446C53E0EB826550CE388C1CEA7BCB3BF1694D8A945A2CEA95B22255F9259941C22BFCBC8C857CBBFBC0EE840F98703BF609B08C68E99C605FC00D66D90A8F5F8D38D43C88F7ABDBB28AC04694A0B867337F06D4F04F6F5AFBFAB8ECE75534D7F7D17780E12464AAF9599EFBCA6C54177437AB9EC8E073C6D
dh1024_g = 2
# the new dh params
dh2048_p = 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
dh2048_g = 2
# is_prime(dh2048_p) -> True
order = dh2048_p - 1
factors = [2]
print "2 divides the order"
# let's try to factorize the order by trial divisions
def find_factors(number):
factors = []
# use different techniques to get primes, dunno which is faster
index = 0
for prime in Primes():
if Mod(number, prime) == 0:
print prime, "divides the order"
factors.append(prime)
if index == 10000:
print "tested up to prime", prime, "so far"
index = 0
else:
index += 1
return factors
factors += find_factors(order / 2)

It has been running for a while now (up to 82018837, a 27bits number) and nothing has been found so far...

The thing is, a Pohlig-Hellman attack is do-able as long as you can compute the discrete log modulo each factors. There is no notion of "small enough factor" defined without a threat model. This backdoor is not gonna be usable by small players obviously, but by bigger players? By state-sized attackers? Who knows...

EDIT: I forgot order/2 could be a prime as well. But nope.

I was watching this excellent video on the birth of elliptic curves by Dan Boneh, and I felt like the explanation of Diffie-Hellman (DH) felt short. In the video, Dan goes on to explain the typical DH key exchange:

Alice and Bob each choose a public point \(g\) and a public modulus \(N\).

By the way. If you ever heard of "DH-1024" or some big number associated to Diffie-Hellman, that was probably the bitsize of this public modulus \(N\).

The exchange then goes like this:

Alice generates her private key \(a\) and sends her public key \(g^a\pmod{N}\) to Bob.

Bob generates his own private key \(b\) and sends his public key \(g^b\pmod{N}\) to Alice.

They can both generate their shared key by doing either \((g^b)^a \pmod{N}\) for Alice, or \((g^a)^b \pmod{N}\) for Bob.

Dan then explains why this is secure: because given \((g, g^a, g^b)\) (the only values an eavesdropper can observe from this exchange) it's very hard to compute \(g^{ab}\), and this is called the Computational Diffie-Hellman problem (CDH).

But this doesn't really explain how the scheme works. You could wonder: but why doesn't the attacker do the exact same thing Bob and alice just did? He could just iterate the powers of \(g\) until \(g^a\) or \(g^b\) is found, right?

A key exchange with hash functions

Let's replace the exponentiation by a hash function. Don't worry I'll explain:

\(g\) will be our public input and \(h\) will be our hash function (e.g. sha256). One more thing: \(h^{3}(g)\) translates to \(h(h(h(g)))\).

So now our key exchange looks like this:

Alice generates an integer \(a\) large enough and compute \(a\) iteration of the hash function \(h\) over \(g\), then sends it to Bob.

Bob does the same with an integer \(b\) and sends \(h^b(g)\) to Alice (exact same thing that Alice did, different phrasing.)

They both compute the share private key by doing either \((h^b(g))^a\) for Alice, or \((h^a(g))^b\) for Bob.

So if you understood the last part: Alice and Bob both iterated the hash functions on the starting input \(g\) a number of \(a+b\) times. If Alice's public key was \(h(h(g))\) and Bob's public key was \(h(h(h(g)))\) then they both end up computing \(h(h(h(h(h(g)))))\).

That seems to work. But how is this scheme secure?

You're right, it is not. The attacker can just hash \(g\) over and over until he finds either Alice's or Bob's public key.

So let's ask ourselves this question: how could we make it secure?

If Bob or Alice had a way to compute \(h^c(x)\) without computing every single hash (\(c\) hash computations) then he or she would take way less time to compute their public key than an attacker would take to retrieve it.

Back to our discrete logarithm in Finite groups

This makes it easier to understand how the normal DH exchange in finite groups is secure.

The usual assumptions we want for DH to works were nicely summed up in Boneh's talk

The point of view here is that discrete log is difficult AND CDH holds.

Another way to see this, is to see that we have algorithm to quickly calculate \(g^c \pmod{n}\) without having to iterate through every integers before \(c\).

To be more accurate: the algorithms we have to quickly exponentiate numbers in finite groups are way faster than the ones we have to compute the discrete logarithm of elements of finite groups. Thanks to these shortcuts the good folks can quickly compute their public keys while the bad folks have to do all the work.

Taken from the SLOTH paper, the current estimated complexities of the best known attacks against MD5 and SHA-1:

Common-prefix collision

Chosen-prefix collision

MD5

2^16

2^39

SHA-1

2^61

2^77

MD5|SHA-1

2^67

2^77

MD5|SHA-1 is a concatenation of the outputs of both hashes on the same input. It is a technique aimed at reducing the efficiency of these attacks, but as you can see it, it is not that efficient.

I never understood why Firefox doesn't display a warning when visiting non-https websites. Maybe it's too soon and there are too many no-tls servers out there and the user would learn to ignore the warning after a while?

A few weeks ago I wrote about testing RSA public keys from the most recent Alexa's top 1 million domains handshake log that you can get on scans.io.

Most public exponents \(e\) were small and so no small private key attack (Boneh and Durfee) should have happened. But I didn't explained why.

Why

The private exponent \(d\) is the inverse of \(e\), that means that \(e * d = 1 \pmod{\varphi(N)}\).

\(\varphi(N)\) is a number almost as big as \(N\) since \(\varphi(N) = (p-1)(q-1)\) in our case. So that our public exponent \(e\) multiplied with something would be equal to \(1\), we would at least need to loop through all the elements of \(\mathbb{Z}_{\varphi(N)}\) at least once.

Or put differently, since \(e > 1 \pmod{\varphi(N)}\), increasing \(e\) over \(\varphi(N)\) will allow us to get a \(1\).

This quick test with Sage shows us that with a small public exponent (like 3, or even 10,000,000), you need to multiply it with a number greater than 1000 bits to reach the end of the group and possibly ending up with a \(1\).

All of this is interesting because in 2000, Boneh and Durfee found out that if the private exponent \(d\) was smaller than a fraction of the modulus \(N\) (the exact bound is \(d < N^{0.292}\)), then the private exponent could be recovered in polynomial time via a lattice attack. What does it mean for the private exponent to be "small" compared to the modulus? Let's get some numbers to get an idea:

That's right, for a 1024 bits modulus that means that the private exponent \(d\) has to be smaller than 300 bits. This is never going to happen if the public exponent used is too small (note that this doesn't necessarely mean that you should use a small public exponent).

Moar testing

So after testing the University of Michigan · Alexa Top 1 Million HTTPS Handshakes, I decided to tackle a much much larger logfile: the University of Michigan · Full IPv4 HTTPS Handshakes. The first one is 6.3GB uncompressed, the second is 279.93GB. Quite a difference! So the first thing to do was to parse all the public keys in search for greater exponents than 1,000,000 (an arbitrary bound that I could have set higher but, as the results showned, was enough).

I only got 10 public exponents with higher values than this bound! And they were all still relatively small (633951833, 16777259, 1065315695, 2102467769, 41777459, 1073741953, 4294967297, 297612713, 603394037, 171529867).

Here's the code I used to parse the log file:

import sys, json, base64
with open(sys.argv[1]) as ff:
for line in ff:
lined = json.loads(line)
if 'tls' not in lined["data"] or 'server_certificates' not in lined["data"]["tls"].keys() or 'parsed' not in lined["data"]["tls"]["server_certificates"]["certificate"]:
continue
server_certificate = lined["data"]["tls"]["server_certificates"]["certificate"]["parsed"]
public_key = server_certificate["subject_key_info"]
signature_algorithm = public_key["key_algorithm"]["name"]
if signature_algorithm == "RSA":
modulus = base64.b64decode(public_key["rsa_public_key"]["modulus"])
e = public_key["rsa_public_key"]["exponent"]
# ignoring small exponent
if e < 1000000:
continue
N = int(modulus.encode('hex'), 16)
print "[",N,",", e,"]"

This is the 3rd post of a series of blogpost on RWC2016. Find the notes from day 1 here.

I'm a bit washed out after three long days of talk. But I'm also sad that this comes to an end :( It was amazing seeing and meeting so many of these huge stars in cryptography. I definitely felt like I was part of something big. Dan Boneh seems like a genuine good guy and the organization was top notch (and the sandwiches amazing).

SGX morning

The morning was filled with talks on SGX, the new Intel technology that could allow for secure VMMs. I didn't really understood these talks as I didn't really know what was SGX. White papers, manual, blogposts and everything else is here.

10:20pm - Practical Attacks on Real World Cryptographic Implementations

practical attacks found as well in TLS on JSSE, Bouncy Castle, ...

exception occurs if padding is wrong, it's caught and the program generates a random. But exception consumes about 20 microseconds! -> timing attacks (case JSSE CVE-2014-411)

invalid curve attack

send invalid point to the server (of small order)

server doesn't check if the point is on the EC

attacker gets information on the discrete log modulo the small order

repeat until you have enough to do a large CRT

they analyzed 8 libraries, found 2 vulnerable

pretty serious attack -> allows you to extract server private keys really easily

works on ECDH, not on ECDHE (but in practice, it depends how long they keep the ephemeral key)

HSM scenarios: keys never leave the HSM

they are good candidates for these kind of "oracle" attacks

they tested and broke Ultimaco HSMs (CVE-2015-6924)

<100 queries to get a key

11:10am - On Deploying Property-Preserving Encryption

tl;dw: how it is to deploy SSE or PPE, and why it's not dead

lots of "proxy" companies that translates your queries to do EDB without re-teaching stuff to people (there was a good slide on that that I missed, if someone has it)

searchable symmetric encryption (SSE): you just replace words by token

threat model is different, clients don't care if they hold both the indexes and the keys

two kinds of order preserving encryption (OPE):

stateless OPE (deterministic -> unclear security)

interactive OPE (stateful)

talks about how hard it is to deploy a stateful scheme

many leakage-abused attacks on PPE

crypto researcher on PPE: "it's over!", but the cost and legacy are so that PPE will still be used in the future

I think the point is that there is nothing practical that is better than PPE, so rather than using non-encrypted DB... PPE will still hold.

11:30am - Inference Attacks on Property-Preserving Encrypted Databases

tl;dw: PPE is dead, read the paper

analysis have been done and it is known what leaks and cryptanalysis have been done from these information

real data tends to be "non-uniform" and "low entropy", not like assumptions of security proofs

inference attacks:

frequency analysis

sorting attack

Lp-optimization

cumulative attacks

frequency analysis: come on we all know what that is

Lp-optimization: better way of mapping the frequency of auxilliary data and the ciphertexts

sorting attacks: just sort ciphertextxs and your auxiliary data, map them

this fails if there is missing items in the ciphertexts set

cumulative attack improve on this

check page 6 of the paper for explanations on these attacks. All I was expecting from this talk was explanation of the improvements (Lp and cumulative) but they just flied through them (fortunately they seem to be pretty easy to understand in the paper). Other than that, nothing new that you can't read from their paper.

2:00pm - Cache Attacks on the Cloud

tl;dw: cache attacks can work, maybe

hypervisor (VMM) ensures isolation through virtualization

VMs might feel each other's load on some low-level resources -> potential side channels

covert channel in the cloud?

LLC is cross core (L3 cache)

cache attacks

prime+probe

priming: find eviction set: memory line that when loaded to cache L3 will occupy a line we want to monitor

probing: when trying to access the memory line again, if it's fast that means no one has used the L3 cache line

to get crypto keys from that you need to detect key-dependent cache accesses

for RSA check timing and number of times the cache is accessed -> multiplications

for AES detect the lookup table access in the last round (??)

cross-VM cache attacks are realistic?

attack 1 (can't remember) (hu)

co-location: detect if they are on the same machine (dropbox) [RTS09]

they tried the same on AWS EC2, too hard now (hu)

new technique: LLC Cache accesses (hu)

new technique: memory bus contention [xww15, vzrs15]

once they knew they were on the same machine through colocation what to target?

libgcrypt's RSA use CRT, sliding window exponentiation and message blinding (see end of my paper to see explanation of message blinding)

conclusion:

cache attacks in public cloud work

but still noise and colocation problem

open problem: countermeasures?

what about non-crypto code?

Why didn't they talk of flush+reload and others?

2:30am - Practicing Oblivious Access on Cloud Storage: the Gap, the Fallacy, and the New Way Forward

Oblivious RAM, he doesn't want to explain how it works

how close is ORAM to practice?

implemented 4 different ORAM system from the litterature and got some results from it

CURIOUS, what they made from these research, is open-source. It's made in Java... such sadness.

Didn't get much from this talk. I know this is "real world" crypto but a better intro on ORAM would have been nicer, also where does ORAM stands in all the solutions we already have (fortunately the previous talk had a slide on that already). Also, I only read about it in FHE papers/presentations, but there was no mention of FHE in this talk :( well... no mention of FHE at all in this convention. Such sadness.

From their paper:

An Oblivious RAM scheme is a trusted mechanism on a client, which helps an application or the user access the untrusted cloud storage. For each read or write operation the user wants to perform on her cloud-side data, the mechanism converts it into a sequence of operations executed by the storage server. The design of the ORAM ensures that for any two sequences of requests (of the same length), the distributions of the resulting sequences of operations are indis-tinguishable to the cloud storage. Existing ORAM schemes typically fall into one of the following categories: (1) layered (also called hierarchical), (2) partition-based, (3) tree-based; and (4) large-message ORAMs.

2:50pm Replacing Weary Crypto: Upgrading the I2P network with stronger primitives

tl;dw: the i2p protocol

i2p is like Tor? both started around 2003, both using onion routing, both vulnerable to traffic confirmation attacks, etc...

but Tor is ~centralized, i2p is ~decentralized

tor use an asymmetric design, i2p is symmetric (woot?)

in i2p traffic works in circle (responses comes from another path)

so twice as many nodes are exposed

but you can only see one direction

this difference with Tor hasn't really been researched

...

4:20pm - New developments in BREACH

tl;dw: BREACH is back

But first, what is BREACH/CRIME?

This talk was a surprise talk, apparently to replace a canceled one?

original BREACH attack introduced at blackhat USA 2013

compression/encryption attack (similar to CRIME)

crime was attaking the request, breach attack the response

based on the fact that tls leaks length

the https server compresses responses with gzip

inject content in victim when he uses http

the content injected is a script that queries the https server

attack is still not mitigated but now we use block cipher so it's OK

extending the BREACH attack:

attack noisy endpoints

attack block ciphers

optimized

no papers?

aes-128 is vulnerable

mitigation proposed:

google is introducing some randomness in their responsness (not really working)

facebook is trying to generate a mask XORed to the CSRF token (but CSRF tokens are not the only secrets)

they will demo that at blackhat asia 2016 in Singapore

4:40pm - Lucky Microseconds: A Timing Attack on Amazon's s2n Implementation of TLS